3. THE PRIMARY UNCERTAINTIES

Measurement errors in m0V are no longer the
limiting factor that determines the
accuracy when using the GCLF for distance determinations. Typical
internal uncertainties are now 0.1 - 0.2 mag, and can be further reduced
by averaging several galaxies
together. The uncertainties introduced by the following questions
generally dominate the calculation.

How secure are the "local" calibrators?

How universal is the GCLF?

How important is interstellar extinction?

What is the appropriate velocity for the calculation of the Hubble
constant?

While most of these are the same questions that were with us 20 and 30
years ago (see
Sandage 1968,
de Vaucouleurs 1970,
Hanes 1979),
the recent increase in the number of
high quality GCLFs and the new capabilities provided by HST are making
it possible to resolve many of these questions. For example, rather than
relying on the Milky Way
and M31 for our calibration it is now possible to use
Cepheid measurements to directly
calibrate the ellipticals in the Virgo cluster.

Until recently, the globular cluster systems of the Milky Way and M31
provided the primary calibration for GCLF distance estimates. While a
few other local galaxies have
also been measured (M33, M81, NGC 4565, NGC 4594; see H96), the data is
relatively
poor and the errors roughly twice as large, so most studies have not
used them.
Secker (1992) and
Sandage & Tammann (1995)
have recently reexamined the use of M31 and the
Milky Way as the local calibrator. Secker finds that
M0V is roughly 0.2 mag brighter in
M31 (M0V = -7.51 ± 0.15
mag) compared to the
Milky Way (M0V = -7.29 ± 0.13 mag).
Sandage and Tammann (1995),
adopting an RR Lyrae calibration for the Milky Way
that is roughly 0.25 mag brighter than in general use, find that M0V is nearly identical
for M31 (M0V = -7.70 ± 0.20
mag) and the
Milky Way (M0V = -7.60 ± 0.11 mag). The
mean of all four values is M0V = -7.52 ± 0.17
mag where the uncertainty is the scatter
in the four values, which provides the best estimate of the accuracy in
this case since systematic effects (i.e. the RR Lyrae distance
scale) dominate the results.

The use of local spirals to calibrate distant ellipticals has always
been one of the
primary weaknesses of the GCLF method for measuring distances, as will
be discussed in the section on universality
(Section 3.2.2). A practical concern is that faint
GCs are more difficult to identify in nearby spirals since they can be
confused with open clusters, and
the smaller number and larger angular extent makes them more difficult
to differentiate from background galaxies.
Reed et al. (1992)
suggests that incompleteness at the faint
end may be the reason the value of M0V in M31
appears to be slightly higher, and the
value of slightly lower,
than the comparable values for the Milky Way.

It is now possible to replace this comparison of "apples and oranges"
with a direct
calibration of elliptical galaxies using measurements of Cepheid
variables in the Virgo cluster galaxies. The primary uncertainty with
this method is the possibility that the
galaxy with Cepheid measurements might be in the front or back of the
cluster. It is
therefore important to observe several galaxies in the same cluster. At
present there are
five galaxies in the Virgo cluster with distance determinations using
Cepheid variables (NGC 4321 -
Ferrarese et al. 1996
find m - M = 31.04 ± 0.17 mag; NGC 4496A -
Saha et al. 1996a
find m - M = 31.03 ± 0.14 mag; NGC 4536 -
Saha et al. 1996b
find m - M = 31.23 ± 0.05 mag; NGC 4571 -
Pierce et al. 1994
find m - M = 30.87 mag; and NGC 4639 -
Sandage et al. 1996
find m - M = 32.00 ± 0.23 mag). The value for NGC 4639
is nearly a magnitude higher than the other values. We therefore remove
the high and
low values to obtain our estimate for the Virgo cluster distance modulus
of m - M = 31.10 ± 0.14 mag (where the error estimate is the mean
of the value for three galaxies
[underestimate] and the value for five galaxies [overestimate]). Using
the value of mV =
23.89 ± 0.09 mag from Section 2.3, and
an uncertainty of 0.20 mag for the Cepheid zeropoint
(Freedman et al. 1994)
yields our final value of M0V = -7.21 ± 0.26
mag. Keeping all five
Cepheid measurements would result in a value of
M0V which is 0.13 mag brighter. The
dominant uncertainty is due to the Cepheid distance determination for
the Virgo cluster
and the Cepheid zeropoint, both of which should improve in the near future.

Hence, the value of M0V derived from the Milky Way
and M31 (i.e., M0V = -7.52 mag) is
0.31 mag brighter than our estimate of M0V for
bright ellipticals in the Virgo cluster. This
should be considered a tentative result due to the large
uncertainties. It is interesting
to note, however, that the value is similar to the what has been
advocated by
Fleming et al. (1995) and
Ashman, Conti, & Zepf
(1995).

In 1958, Sandage wrote "We are, therefore, forced to the conclusion that
globular clusters
are very poor distance indicators because, at least with the present
data, the absolute
magnitude apparently vary from galaxy to galaxy". This was based on a
comparison between the Milky Way and M31.
van den Bergh (1975)
came to essentially the same
conclusion. As the data improved, these discrepancies went away, and it
became more
and more evident that the GCLF was actually quite a good distance
indicator. For example,
Sandage & Tammann (1995)
find M0V for the Milky Way and M31 agree to
within 0.1 mag, well within the statistical uncertainty, and argue that
the GCLF is nearly universal.

Do bright and faint galaxies have the same value of
M0V? H96 treats this in some detail,
showing that dwarf ellipticals have values of M0V
which are roughly 0.3 mag fainter than
bright ellipticals. One possibility is that fainter clusters can survive
in these systems due
to the weaker tidal fields. This would predict that values of
would be larger. While
the value of for a
composite of 8 dwarf ellipticals in Virgo cluster observed by
Durrell et al. (1996)
is the highest measured value in
Table 1 from H96
( = 1.48 ± 0.20) the
large uncertainty due to the small number of clusters makes this very
uncertain. The new HST snapshot survey of 30 dwarf ellipticals
(Ferguson et al. 1996)
should improve the situation.

The difference in M0V for faint galaxies is not
actually very important for the distance
scale problem, since we are most interested in the bright ellipticals
with large populations
of GCs needed to provide small statistical uncertainties. The study of
the GCLF in dwarf
galaxies will primarily be useful for understanding the formation and
evolution of GCs.

The question of whether spiral and elliptical galaxies have the same
GCLF is important since, until recently, the local calibrators were all
spirals while the distant galaxies were all
ellipticals. This circumstance arose because a more accurate value of
M0V can be measured
for a giant elliptical which has more GCs, fewer point-like sources to
be confused with the GCs (e.g. open clusters, HII regions, bright
stars), and negligible internal extinction.
Unfortunately, there are no giant ellipticals in the local group. The
measurement of
Cepheid variables in galaxies as far as the Virgo cluster is now
circumventing this problem
(see Section 3.1), but at the price of making the
GCLF a secondary distance indicator. The
question of universality versus Hubble type is still very important in
order to provide a cross check on the Cepheid scale, as well as to gain
a better understanding of how GCs form and evolve.

As the precision of observations has increased it has become clear that
many galaxies
have bi- or multi-modal color distributions. Recent improvements in
spectral synthesis models (e.g.Bruzual & Charlot 1993,
1996,
Worthey 1994,
Fritze-v.Alvensleben &
Gerhard 1994)
now make it possible to interpret these effects in terms of metallicity
and/or age differences.
Ashman, Conti, & Zepf (1995)
have employed the
Worthy (1994)
stellar evolution models to show that if spirals and ellipticals have
the same mass function
(an assumption that needs to be carefully checked, since there are a
variety of physical mechanisms that could modify the GC mass function;
see Section 4), then spirals should have
values of M0V which are roughly 0.2 mag brighter
than ellipticals due to differences in the
mean metallicity of the globular clusters. They also show that the I
band is less sensitive to metallicity effects.

The discovery of candidate young globular clusters in both merger
remnants (e.g.Holtzman et al. 1992,
Whitmore et al. 1993,
Whitmore & Schweizer 1995)
and in starburst galaxies (e.g.Conti & Vacca 1994,
Hunter, O'Connell, &
Gallagher 1994,
and Meurer et al. 1995)
raises the question of whether M0V might be
affected by the presence of young and intermediate-age clusters for some
galaxies. While very young clusters are
easy to identify, since they are up to 6 magnitudes brighter in
M0V and 0.4 to 1.0 mag bluer in V-I,
Whitmore et al. (1996)
show that intermediate-age clusters are more difficult
to isolate. At an age of 1 - 2 Gyr the clusters have roughly the same
colors as old globular clusters, based on the
Bruzual & Charlot (1996)
models, although they are 1 - 2
mag brighter. Older clusters are redder, and either slightly brighter (2
- 6 Gyr) or slightly
fainter (> 6 Gyr) than the old population. Intermediate-age clusters
may therefore lead
to an estimate for M0V which is either slightly
too large or too small. It may be possible to
minimize this potential problem by selecting galaxies that do not have
evidence of recent
merger activity (e.g. shells and loops), or by using only the
blue clusters to determine M0V.

There are now enough galaxies with well determined GCLFs to begin to ask
whether the environment and the position within the galaxy (i.e.
the local environment) can affect the GCLF.

Blakeslee and Tonry (1996),
using only six galaxies, find tentative evidence for a
correlation between MV and environment. Using the group
velocity dispersion as an indicator
of environment they find that M0V is fainter by
about 0.5 mag in the Coma cluster than
in the local group. However, the correlation is quite dependent on the
observations of NGC 4881 in the Coma cluster
(Baum et al. 1995)
which are ambiguous (see Section 2.2). In
addition, even if the correlation turns out to be real it is quite
possible that a difference
in M0V for different Hubble types is the cause
rather than an environmental effect, since
the galaxies in the low velocity dispersion environment are all spirals
while the other
galaxies are ellipticals. Finally, the opposite trend appears to be
present for the dwarf
ellipticals in the field compared to the dwarf ellipticals in the
Virgo cluster (H96).

A large number of studies have concluded that M0V
is not a function of position in a galaxy (e.g.Grillmair, Pritchet, & van
den Bergh 1986,
van den Bergh 1992,
McLauglin, Harris, & Hanes
1994,
Harris & Pudritz 1994,
and Forbes 1996a).
However, a recent paper by
van den Bergh (1996)
indicates that the luminosity distribution of globular
clusters in the Milky Way depends on cluster size, which is known to
correlate with Galactocentric distance. In addition, the clusters in the
outer halo (> 10 kpc) with red
horizontal branches are fainter than normal clusters by a factor of
10. Many of these are
the anomalous clusters such as Pal 1, Pal 12, and Ter 7 which are both
very extended and
very faint, and would not be picked up by most studies of GC's in
external galaxies. It
will be important to examine the more recent HST datasets to search for
possible trends in ellipticals, but based on earlier results the effects
on M0V are likely to be quite small.

Extinction from dust in the Milky Way (galactic extinction), and dust in
other galaxies (internal extinction), results in relatively minor
uncertainties. For example,
Sandage & Tammann (1995)
advocate a value of AV = 0.00 mag for the galactic extinction
in M87 while several other studies use a value from
Burstein and Heiles (1984)
of AV 0.06
mag (e.g.Whitmore et al. 1995).
The situation is similar for the Fornax cluster, where
values of AV = 0.00 mag, 0.05 mag, and 0.10 mag are commonly
used.

Effects of internal extinction are generally negligible since elliptical
galaxies, containing very little gas and dust, are used for most GCLF
distance determinations. However, for
the Milky Way GCs typical values of the color excess, E(B - V), are 0.3
mag
(Sandage & Tammann 1995),
and 0.1 mag for M31
(Reed, Harris, & Harris
1992).
Internal extinction is potentially important if these galaxies are used
to set the zeropoint. If we use galaxies
with Cepheid variables to calibrate our zero point the uncertainty due
to extinction is quite small. For example,
Ferrarese et al. (1996)
estimate a total color excess E(B - V) = 0.10 mag for the Cepheid
measurements in M100, and a combined uncertainty of 0.02
mag in the error budget for absorption in M100 and the LMC (used to define the Cepheid zeropoint).

A determination of the Hubble constant requires an estimate of the
distance and of the appropriate "expansion" velocity of a
galaxy. Globular clusters share a problem common
to several distance indicators, namely they only provide accurate
distance estimates out
to a few thousand km s-1. Since peculiar velocities of a few
hundred km s-1 are common
for clusters of galaxies (e.g.Lauer & Postman 1994),
this can introduce substantial
uncertainties in our estimate of the Hubble constant. For example, a
typical estimate for
the Virgocentric infall velocity is 200 km s-1
(Han & Mould 1990).
This represents an
15% correction to the
Virgo velocity. The spread in the different measurements
( 70
km s-1) represents a 5% uncertainty in H0.

One approach to circumvent this problem is to use HST to push out to
larger distances.
Baum et al. (1995)
has shown that it is difficult but not impossible to reach the turnover
of the GCLF at the distance of the Coma cluster
( 7000 km
s-1). Another technique is
to offset from a nearby cluster to a more distant cluster, assuming the
relative distances of the two clusters are well known (e.g.Freedman et al. 1994).
This appears to be the
case between the Virgo and Coma clusters, where estimates of
Coma-Virgo(m - M) =
3.71 ± 0.05 mag
(van den Bergh 1992,
based on 12 studies),
Coma-Virgo(m - M)
= 3.71 ± 0.03 mag
(de Vaucouleurs 1993,
based on 12 studies, with 7 overlapping
van den Bergh 1992),
and Coma-Virgo(m - M)
= 3.80 ± 0.17 mag
(Jerjen & Tammann 1993,
based on five studies) are in good agreement. A related technique is to
use the relative distance
between a single cluster and several more distant clusters, so that any
peculiar motions of the distant clusters tend to average out.
Jerjen and Tammann (1993)
use this technique to estimate the "Machian" velocity of the Virgo cluster based on 15 clusters. The main
shortcoming of this technique is that the number of galaxy clusters with
accurate relative distance is limited, so adding more clusters may
simply add noise in some cases.

It is generally assumed that all the galaxies in a cluster are at the
same distance, and the systemic velocity of the cluster can therefore be
used as the appropriate velocity for the calculation of the Hubble
constant. However, the galaxy may actually be in the front
or back of the cluster. For a system as large and ill-defined as the
spiral galaxies in the
Virgo cluster (i.e. with a "radius" in the
range 3° - 6°), this may result in a 5 - 10%
error in the true distance. In addition, for some studies there may be a
tendency to select
galaxies which are slightly nearer (e.g. Cepheids), since it is
easier to resolve the stars in
this case. It will therefore be important to observe a number of
galaxies and a number of different groups and clusters.

One way to solve this problem is to observe the dominant D or cD galaxy
in the cluster, since there is good evidence that these galaxies reside
very near the center of the cluster.
For example, the position at the center of the extended x-ray halo in
the Virgo cluster is good evidence that M87 is at the
bottom of the potential well. Another method is to use
a very compact cluster such as Fornax. In this case the radius of the
cluster is roughly
1°, implying a maximum line-of-sight excursion of about 2% if the
cluster is roughly spherical.

As discussed in Section 2.3, the
combination of the intrinsic dispersion in the value of
M0V and
the measurement uncertainty results in internal uncertainties of
0.2 mag. Data for
several galaxies can be averaged together reducing the uncertainty still
further. Hence, the external errors are generally the limiting factors
for distance determinations using the GCLF.

Table 3 provides an estimate of the total error
budget for the following three cases: 1)
using the Milky Way and M31 for calibration, 2) using Cepheids in the
Virgo cluster for calibration, 3) using Cepheids in
several groups and clusters to nail down the zeropoint,
and ten ellipticals in the range 2,000 - 5,000 km s-1 with
good sky coverage to average
out peculiar velocities. See Section 2 and
Section 3 for the discussion of the values in
Table 3. Values in
parenthesis assume ten galaxies have been averaged together. The value
of ± 0.10 mag for
the calibration of Method # 3 assumes the uncertainty in the Cepheid
distance scale will
be reduced by roughly a factor of two in the next few years (e.g.
more observations in the
Virgo and Fornax cluster and a better determination of the
distance to
the LMC). The
table shows that at present, the limiting factors are the zeropoint
calibration, peculiar
velocities, and for Method # 1, the question of universality. These lead
to current uncertainties of
15% in H0. The estimate for method # 3 shows that within a
few years it should be possible to use the GCLF to determine distances
for individual
ellipticals to 10%, and
to determine H0 to about 7%.

Table 3. Error estimates

Method # 1

Method # 2

Method # 3

Timeframe:

Current

Current

Future

Calibration:

Milky Way + M31

Virgo Cepheids

Cepheids

Velocities:

Virgo cluster

Virgo cluster

10 distant galaxies

Distance Estimates

Intrinsic Scatter:

0.12 mag (0.04 mag)

0.12 mag (0.04 mag)

0.12 mag (0.04 mag)

Measurement:

0.16 (0.05)

0.16 (0.05)

0.10 (0.03)

Calibration:

0.17

0.24 mag

0.10

Universality:

0.2

-

-

Extinction:

0.05

0.05

0.05

totals:

0.33 mag (0.27 mag)

0.32 mag (0.25 mag)

0.19 mag (0.12 mag)

H0 Estimates

Velocities:

0.15 mag

0.15 mag

(0.05 mag) (0.05 mag)

totals:

0.37 mag (0.31 mag)

0.35 mag (0.29 mag)

0.20 mag (0.13 mag)

NOTES TO TABLE 3

(a) Values in parenthesis are for an average of ten galaxies.

(b) Total errors are derived assuming uncertainties add in quadrature.

(c) Universality is not relevant for methods # 2 and # 3 since the
elliptical galaxies are calibrated directly,
rather than using local spirals.

(d) Effects of peculiar velocities are estimated for methods # 1 and #
2 by assuming an uncertainty in the
expansion velocity for the Virgo cluster of 100 km s-1
(approximate difference between
Jerjen & Tammann [1993]
and Huchra [1988];
using a mean velocity from the two studies of 1292 km s-1).